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Categorified symplectic geometry and the classical string

By John C. Baez, Er E. Hoffnung and Christopher L. Rogers


A Lie 2-algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures similar to those of a Lie algebra, but where the usual laws hold only up to isomorphism. It is well known that given a manifold equipped with a symplectic 2-form, the Poisson bracket gives rise to a Lie algebra of observables. Multisymplectic geometry generalizes the classical mechanics of point particles to n-dimensional field theories, decribing such a theory in terms of a ‘phase space ’ that is a manifold equipped with a closed nondegenerate (n + 1)-form. Here, given a manifold with a closed nondegenerate 3-form, we construct a Lie 2-algebra of observables. We then describe how this Lie 2-algebra can be used to describe dynamics in classical bosonic string theory.

Year: 2010
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